Use with the (group-theory) tag. The tag "finite-groups" refers to questions asked in the field of Group Theory which, in particular, focus on the groups of finite order.

learn more… | top users | synonyms

16
votes
1answer
132 views

Do there exist simple groups of order $n(n+1)$ for some integer $n>1$?

In the comments to this question, it was remarked that the question of whether a finite group of order $p(p+1)$ is necessarily not simple becomes fairly interesting if we do not assume that $p$ is a ...
0
votes
1answer
31 views

Why does Lagrange come into play here? [duplicate]

I am given this problem, and the solution reasons differently from me and obscurely, to me Let $G$ have order $30$. Show that it is not simple. The solution is apparently So, what I don't ...
2
votes
3answers
31 views

Sylow theorem; Sylow $III$

Just curious, I have't seen any statement that answers this so far. And I'm not at the stage of being able to or given a proof of the theorem so I am not sure. Sylow $III$; If $G$ has order ...
1
vote
2answers
31 views

Are the normalizers of Sylow p-subgroups isomorphic?

Let $G$ be a finite group and $A,B \in \text{Syl}_p G$, for some prime $p$. Is it always true that the normalizers $N_G(A)\cong N_G(B)$? I just need a hint to get started, because I don't know where ...
1
vote
1answer
87 views

Group of order $p^2+p $ is not simple [duplicate]

Can some one please give me a hint to prove that every group of order $p^2+p$ is not simple?
1
vote
1answer
12 views

first homology group with coefficients in divisible group

I had (perhaps very elementary) doubt in the understanding of the computation of first homology group of a finite group over a divisible group. Let $\pi$ be a finite group of order $n$ and $D$ be a ...
0
votes
2answers
24 views

The point of a group-theoretic Chinese Remainder Theorem?

It states that for coprime $m,n$ nonzero integers, $C_{mn} \cong C_m \times C_n$. However, I know a theorem that says Cyclic groups with the same order are isomorphic. So $C_{mn} \cong C_m ...
0
votes
0answers
34 views

Non-abelian groups of order $50$

Are there non-abelian groups of order $50$? If so, how many elements of order $2$ can they have? Such a group would have a unique $5$-Sylow-group, so it would have at least $25$ elements not of order ...
1
vote
1answer
17 views

Crossed homomorphism from semi-direct product: confusion in definition

(Ref: this) Let $\pi \times_{\varphi} G$ be semi-direct product in which $G$ is normal and $\pi$ is complement. Let $\omega$ be another complement of $G$ in above semi-direct product (so $\pi ...
1
vote
0answers
24 views

Ordering of elements in the base of a group

In section 4.6.7 of HANDBOOK OF COMPUTATIONAL GROUP THEORY, the authors use an ordering $\prec$ for the elements in a coset. That ordering, $\prec$, was defined in section 4.6 as follows. Throughout ...
0
votes
1answer
23 views

Website or book with Hasse diagrams of subgroups

I need to look at Hasse diagrams of very many groups, especially high powers of small symmetric groups. Is there any place where I could look them up? Calculating them myself would be a huge amount of ...
2
votes
1answer
60 views

What does being Abelian have to do at all with the proof?

I don't understand why the proof needs to consider cases that $G$ is Abelian and non-Abelian. If $|G|=p^n$ where $n>1$ then show that $G$ cannot be simple. It uses the theorem If $G$ is a ...
4
votes
3answers
55 views

Abelian finite group [duplicate]

This is a (maybe be simple) problem from Group Theory, but being a beginner, I am unable to take even a first step forward. Let $G$ be a finite group whose order is not divisible by $3$.Suppose that ...
2
votes
2answers
27 views

Does “order of a subgroup $n$” mean “there is an element of order $n$ in $G$”?

I am not sure about this. I understand it when it is cyclic. But if not stated so, I cannot reason as to why.($G$ here I assume finite) $G$ is a group with some subgroup $H$. Then, if $|H|=n$ then ...
1
vote
1answer
24 views

Permutation and Linear Representation of Finite Group

By a permutation representation of a finite group $G$, we mean a homomorphism from $G$ to $S_n$, the (full) permutation group on $n$ letters. By a linear representation of a finite group $G$, we mean ...
2
votes
1answer
45 views

Representation of Sylow which does not extend

Let $H$ be a subgroup of a finite group $G$ and $\rho$ a representation of $G$ such that the restriction of $\rho$ to $H$ is invariant under conjugation in $G$, in the sense that its character is ...
1
vote
2answers
14 views

Showing that the primary component $G_p$ is a subgroup of $G$

For a finite abelian group $G$ and a prime number $p$ with $p \mid |G|$, we define $G_p$ as the subset of $G$ that contains all elements of $G$ with order $p^k$ for a $k \in \mathbb{N}_0$. We call ...
1
vote
2answers
48 views

Show that $x^{n-1} =1$ for all non-zero element in a field

Question: Show that $x^{n-1} =1$ for all non-zero element in a field Let F be a finite field of order n. Show that $x^{n-1}=1$ for all non-zero $x \in F$. We have $\left | F \right |=n.$ ...
1
vote
0answers
18 views

Groups and queues and stacks

As I review my elementary CS material to prepare for an interview I cannot help but think that I missed a key connection when studying this prior: I think I missed the relationship between operations ...
1
vote
1answer
94 views

About subsets of finite groups with $A^{-1}A=G$ or $AA^{-1}=G$?

Regarding to the problems Does $A^{-1}A=G$ imply that $AA^{-1}=G$? and Is it true that if $|A|>\frac{|G|}{2}$ then $A^{-1}A=AA^{-1}=G$?, we are looking for some subsets $A$ of $G$ with $\lfloor ...
4
votes
3answers
68 views

Show that the permutation $(1 \space 2 \space 3)$ can not be a cube of any element of $S_n.$

Here is my try: If there exists $a \in S_n$ such that $a^3=(1 \space 2 \space 3)$, then $a^9=e$ where $e$ is identity in $S_n$. Then $o(a)=9$. I don't know how to proceed further. Can anyone ...
1
vote
1answer
34 views

Study of specific Quotients of a $p$-group in MAGMA

In Magma, I wanted to do a program, but since I was getting errors in "basic" commands, I am in trouble. Here I will describe a problem to be done in MAGMA, then its interpretation in MAGMA. I would ...
0
votes
0answers
16 views

Primary and Secondary invariants for finite groups

For a finite group G and complex representation V of degree n, I would like to know the precise definition of Primary invariants. Does any set of n algebraically independent homogeneous invariants ...
1
vote
1answer
25 views

$p$-Groups and the Cauchy theorem

Here is the definition of a $p$-group $p$ is prime. A $p$-group is a group $G$ such that every element has an order of a power of $p$. So let me check my understanding, every element of $G$ has ...
0
votes
0answers
15 views

Conjugacy Class Equation for $\mathbb{Z_{25}}$

I'm suposed to find the conjugacy class equation for $\mathbb{Z_{25}}$. Since $\mathbb{Z_{25}}$ is Abelian, that means that $gxg^{-1}=xgg^{-1}=x$ so $Z(\mathbb{Z_{25}})=\mathbb{Z_{25}}$ and every ...
2
votes
2answers
31 views

Kazhdan-Lusztig polynomials for the longest element in finite Coxeter groups

Suppose $W$ is a finite Coxeter group and that $w_\circ$ is the longest element. I want to prove that $P_{x,w_\circ}=1$ for all $x \in W$. I know from Corollary 7.14 in the book Humphreys p. 167 ...
2
votes
1answer
27 views

Relation between conjugate subgroups and the subgroup generated by them

Let $G$ be a group with $H\leq G$. Suppose that $g\in G$ and $y\in \langle H, gHg^{-1}\rangle$. If $yHy^{-1}$ and $gHg^{-1}$ are conjugate in $\langle yHy^{-1}, gHg^{-1}\rangle$, then $H$ and ...
1
vote
0answers
46 views

What is the name of this finite group of order 36

In my research we find a finite group of order 36, which satisfies the following generating relations \begin{equation} \langle g_{12}~,~g_3 ~|~g_{12}^{12}=e=g_3^3, ~~~g_{12}~g_3=g_3^2~g_{12}\rangle ...
0
votes
3answers
55 views

Group of order $p^2$ [duplicate]

Let $G$ be a group of order $p^2$ where $p$ is a prime. I want to show that either $G$ is cyclic or is the product of two cyclic groups of order $p$. After some work, the problem reduces to showing ...
1
vote
1answer
23 views

Showing a translation group is a normal subgroup of an affine group

Let V be an n-dim vector space over the field $\mathbb{F}.$ $A \in GL\left ( n,\mathbb{F} \right )$ and $v \in V$ Define the affine transformation $t_{A,v}$: $V\rightarrow V$ $x \mapsto xA+v$ ...
0
votes
0answers
42 views

Proof of the fixed point theorem in group action.

$\textbf{Fixed point theorem:}$ Let $G$ be a $p$-group and let $S$ be a finite set on which $G$ operates. If the order of $S$ is not divisible by $p$, there is a fixed point for the operation of $G$ ...
1
vote
0answers
30 views

How does the Whitehead quadratic functor act on morphisms?

I am trying to understand the universal quadratic functor $\Gamma$ that appears on the Whitehead exact sequence (J. Whitehead, A certain exact sequence, Annals of Mathematics 52 (1950)) particularly ...
0
votes
0answers
19 views

Showing the affine transformation is well-defined

Let V be an n-dim vector space over the field $\mathbb{F}.$ $A \in GL\left ( n,\mathbb{F} \right )$ and $v \in V$ Define the affine transformation $t_{A,v}$: $V\rightarrow V$ $x \mapsto xA+v$ ...
1
vote
0answers
27 views

PGL(2,c) is isomorphic to PSL(2,c)

Definition: Projective special linear group $PSL\left ( n,\mathbb{F} \right )=\frac{SL\left ( n,\mathbb{F} \right )} {\left ( Z\left ( GL\left ( n,\mathbb{F} \right ) \right )\cap SL\left ( ...
1
vote
1answer
23 views

Language used in projective linear group

In lectures and text on topic of projective linear group, I hear and see the word "factor out" or "quotient out" thrown around a lot. What is the word supposed to mean? If this is vague, I can ...
1
vote
1answer
52 views

Fulton and Harris exercise 5.7

I would like to know if I'm on the right track: We're trying to derive the character table for $PGL(2,q=p^k)$ from $GL(2,q)$. The table for $GL$ is given in the text. It gives the hint that the ...
1
vote
1answer
30 views

Quotient group with normal subgroup dividing the order of another group [duplicate]

Let G be a group with subgroup H and let $\Omega$ be the set of right cosets of H in G. Show that if G is a group with a subgroup of index n then G has a normal subgroup with index dividing n! ...
2
votes
0answers
38 views

Irreducible components of tensor product representations.

Let $(\rho,V)$ be an irreducible representation of a finite group $G$, and let $W$ be a vector space. Then clearly $(\rho\otimes\text{Id}_{W},V\otimes W)$ is also a representation of $G$. I would like ...
0
votes
1answer
60 views

Rings where $ab=0$ for all elements

Let $R$ be a ring, not necessarily unital, such that $ab=0$ for all $a,b\in R$. Suppose $R$ only has trivial right ideals. Is it true that $R$ has finite order? Are these rings special?
0
votes
0answers
16 views

Inverse of an element in an external direct product

Let $G = \mathbb{Z}_{4}\times S_{5}$ What is the inverse of $\left ( 3,\left ( 1,2 \right )\left ( 3,5 \right ) \right )?$ The inverse of any elements a in $\mathbb{Z}_{4}$ is 4-a. So the inverse of ...
9
votes
0answers
78 views

Restricting irreps of $S_n$ to $D_n$ of order $2 n$

I would like to know how to restrict the irreps of the symmetric group $S_n$ to the dihedral group $D_n$ of order $2 n$. We know that $D_n < S_n$. Symmetric group $S_n$ Due to Hardy and Ramanujan ...
0
votes
0answers
20 views

Looking for conditions on generator sets of the form $\{a,b \}$ and $\{a,b,c\}$ on the group $\mathbb{Z}_2^3 \rtimes S_3$

Let $G$ be the group $\mathbb{Z}_2^3 \rtimes S_3$ with the natural action of $S_3$ on the coordinates of $\mathbb{Z}_2^3$. I want to know if there are subsets of $G$ of two elements or also 3 elements ...
1
vote
1answer
32 views

Why is Frobenius norm related to the inner product of characters?

This is a continuation of my question asked here. I am going through Normal Subgroup Reconstruction and Quantum Computation Using Group Representations. In Definition 2, the authors start with the ...
1
vote
1answer
19 views

Probability of measuring the label of representation in quantum Fourier transformaton

I am going through Normal Subgroup Reconstruction and Quantum Computation Using Group Representations. In Definition 2, the authors start with the following function. $$ f : G \to \mathbb{C} $$ Then ...
1
vote
1answer
30 views

Why is the sum of irreducible representations nonzero only when the irreducible representation is trivial?

I am going through Normal Subgroup Reconstruction and Quantum Computation Using Group Representations. In section 3, the authors discuss the probability of measuring the irreducible representation ...
1
vote
0answers
60 views

When can a dihedral group $D_{n}$ of order $2 n$ be a $p$-group?

A $p$-group is a group where the order of every group element is a power of the prime $p$. The presentation of a dihedral group $D_n$ of order $2 n$ is as follows. $$D_n = \langle x, y \mid x^n = y^2 ...
1
vote
1answer
87 views

What is $gnu(18,480)\ $?

Probably, the number of groups of order $18,480$ can only be determined with GAP. But I may be wrong, so the question should not be understood to be purely computational. A better lower bound than my ...
2
votes
1answer
30 views

General property of Fitting series

Let $G$ be a finite solvable group. $F(G)$ (Fitting subgroup) is defined to largest normal nilpotent group contained in $G$. Then $F_2(G)$ is defined to be inverse image of $F(G/F(G))$. i.e ...
0
votes
0answers
56 views

What is the name of the group $\mathbb Z_2\times \mathbb Z_2\times \mathbb Z_2$?

I know, that $\mathbb Z_2\times \mathbb Z_2$ is the Klein four-group. Is there a nice name for $\mathbb Z_2\times \mathbb Z_2\times \mathbb Z_2$ too?
1
vote
1answer
36 views

Fourier transformation of a group

At the beginning of the section 4 of Fast Quantum Fourier Transforms for a Class of Non-abelian Groups, it is said that, ... calculating a Fourier transform for a group $G$ is the same as decomposing ...